A simple regression has been carried out on a set of response variables,

(I!, , . . . , yJ, and a corresponding set of explanatory variables, (xl, . . . . &).

There is just one explanatory variable. The Gaussian linear regression model has been adopted. Adopt a vague prior density for the regression coefficients and the error variance.

(a) Find the joint posterior distribution for the unknown parameters.

(b) Find the predictive distribution and an associated 95 petrent credibility interval for E(f I {x,, . . . , xJ), where y* denotes a new response variable observation predicted from a new value of the explanatory variable, x*.

(c) Find the predictive distribution and an associated 95 percent credibility interval for f.

(d) Why are the lengths of the credibility intervals in parts

(b) and (c)
different?